March 9, 2022
Simple population: \[\large N(t)\]
Age-structured: \[\large N_i(t) = \{N_1(t), N_2(t), ..., N_k(t)\}\]
Spatial structure: \[\large N_i(t) = \{N_1(t), N_2(t), ..., N_k(t)\}\]
The local populations MUST be somehow connected via dispersal.
There must be areas of (near) zero density in between. The “in-between” is referred to as the matrix.
WA sea otters
All sea otters
\(e\) = local probability of extinction
Take away: Even with very LOW probability of extinction, you WILL go extinct.
\(k\) populations, \(t\) time steps
| Pops: | One time step | t steps |
|---|---|---|
| 1: | \(1-e\) | \((1-e)^t\) |
| 2: | \(1-e \times e\) | \((1-e^2)^t\) |
| 3: | \(1-e \times e \times e\) | \((1-e^3)^t\) |
| k: | \(1-e^k\) | \((1-e^k)^t\) |
Metapopulations DRAMATICALLY spread out the risk of extinction!
But still - if ONLY process is extinction … you WILL go extinct (sorry!)
\[P(k,t) = (1-e^k)^t\]
Island-Mainland model
Note - there is also here an important (implicit) assumption that population very quickly hits carrying capacity, so essentially instant saturation.
The mainland is a constant, independent source of potential colonizers. Also known as propagule rain.
(echoes of biogeography).
\[E(N_{t+1}) = N_t - p_e N_t + (K - N_t)\,p_c\]
define proportion of populated patches: \(f_t = E(N_t)/K\), and define equilibirum: \[f^* = f_{t+1} = f_t\]
\[\large f^* = {p_c \over p_c + p_e}\]
Very general metapopulation model: \[\large {df \over dt} = c - e\]
Where \(c\) - colonization rate, \(e\) - extinction rate.
Note: this is similar to \({dN \over dt} = b - d\).
(Highly unrealistic) assumptions!
But - lots of elegant conclusions can be made messing with this model.
\[\large {df \over dt} = c - e\]
Colonization is constant, so proportional to available patches: \[c = p_c(1-f)\]
Extinction is constant, so proportional to occupied patches:
\[e = p_e f\]
so: \[ {df \over dt} = p_c(1-f) - p_ef\]
in words: The rate of change of the occupied patches GROWS in proportion to unoccupied patches and FALLS in proportion with occupied patches.
\[ {df \over dt} = p_c f(1-f) - p_ef\]
Extinction is constant, as before: \[e = p_e f\]
Colonization can only come from occupied patches: \[c = p_c \, f \, (1-f)\]
If no patch is colonized (\(f=0\)), nothing can colonize.
If the population is 100% occupied (\(f = 1\)), there is nothing to colonize.
\[ \large {df \over dt} = p_c f(1-f) - p_ef\]
\[ \large {df \over dt} = p_c (1-f) - p_ef(1-f)\]
Assumes that if you have a lto of neighbors some loose “propagules” will buffer you from extinction.
Equilibrium states:
\[f^* = \begin{cases} p_c/p_e & \text{when} & p_e > p_c \\ 1 & \text{when} & p_e \leq p_c\end{cases}\]
Even with higher extinction rate than colonization rate, there will always be some occupied patches!
\[ \large {df \over dt} = p_c f(1-f) - p_ef(1-f)\]
Only equilibria: 0, if \(p_e > p_c\) or 1, if \(p_e \leq p_c\).
Fundamental conclusions: metapopulation under equilibrium MUST be rare! - either everything colonizes or nothing colonizes.
With rather different predictions! (Nice synthesis - mainly due to Gotelli.)
but how can it be applied with so many unrealistic assumptions?
With case-specific modeling!
It’s hard to do metapopulation studies!
Pinto / northern abalone (Haliotis Kamtschatkana) - overharvested commercially in British Columbia, commercial harvest banned, recovering.
Question: how is “poaching” affecting their recovery?
Used: metapopulation probability of extinction = 0.1 as threshold, corresponding to IUCN definition of Vulnerable.
(from: Hanski, I. 1997. Metapopulation biology. Pp. 69-91. San Diego, USA, Academic Press.)